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Numerische Mathematik

, Volume 133, Issue 4, pp 685–705 | Cite as

Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves

  • Wolfgang Erb
  • Christian KaethnerEmail author
  • Mandy Ahlborg
  • Thorsten M. Buzug
Article

Abstract

Motivated by an application in Magnetic Particle Imaging, we study bivariate Lagrange interpolation at the node points of Lissajous curves. The resulting theory is a generalization of the polynomial interpolation theory developed for a node set known as Padua points. With appropriately defined polynomial spaces, we will show that the node points of non-degenerate Lissajous curves allow unique interpolation and can be used for quadrature rules in the bivariate setting. An explicit formula for the Lagrange polynomials allows to compute the interpolating polynomial with a simple algorithmic scheme. Compared to the already established schemes of the Padua and Xu points, the numerical results for the proposed scheme show similar approximation errors and a similar growth of the Lebesgue constant.

Mathematics Subject Classification

41A05 41A10 42B05 65T40 

Notes

Acknowledgments

The authors gratefully acknowledge the financial support of the German Federal Ministry of Education and Research (BMBF, Grant number 13N11090), the German Research Foundation (DFG, Grant number BU 1436/9-1 and ER 777/1-1), the European Union and the State Schleswig-Holstein (EFRE, grant number 122-10-004).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Wolfgang Erb
    • 1
  • Christian Kaethner
    • 2
    Email author
  • Mandy Ahlborg
    • 2
  • Thorsten M. Buzug
    • 2
  1. 1.Institute of Mathematics, Universität zu LübeckLübeckGermany
  2. 2.Institute of Medical EngineeringUniversität zu LübeckLübeckGermany

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